High-contrast Imaging with Fizeau Interferometry: the Case of Altair^*

The technique of direct imaging offers the possibility of studying the emitted spectra of massive exoplanets that lie at large angular separations from their host stars (e.g., Thalmann et al. 2011; Quanz et al. 2012; Vigan et al. 2015, 2021; Rajan et al. 2017; Greenbaum et al. 2018; Boehle et al. 2019; Nielsen et al. 2019; Hunziker et al. 2020). To overcome the severe planet-to-star contrast, the first generation of direct imaging surveys preferentially targeted younger stars with ages ranging from a few Myr to ≈500 Myr, because massive planets in such systems would still be glowing with the heat of formation—a glow that drops by orders of magnitude as the system ages from ∼1 Myr to ∼1 Gyr (Mordasini et al. 2012; Marleau & Cumming 2013; Bowler 2016).

However, circumstances could also favor the imaging of planets around older stars, if they are particularly close to Earth. Within an arbitrarily chosen 10 pc radius from Earth, there are 68 A- through K-type stars (Reylé et al. 2021). One of the four A-type stars in that sample, Altair, is a 5.130 ± 0.015 pc distant δ Scuti variable (Buzasi et al. 2005; Van Leeuwen 2007). At m_V = 0.8 mag on the Vega scale, it is the 12th-brightest star to the human eye in the night sky (Hoffleit & Warren 1995). The sheer brightness of Altair quickly saturates detectors and makes it a challenging candidate for transit or precise astrometric studies, and it will be technically challenging for the James Webb Space Telescope even with a defocused point-spread function (PSF) and shorter subarray readout times (Beichman et al. 2014; Talens et al. 2017).

The tightest astrometric constraints are from Hipparcos observations over a baseline of 2.7 yr, and are consistent with a single star. The parallax precision of ≈0.6 mas (Perryman et al. 1997; Van Leeuwen 2007) is equivalent to the astrometric semiamplitude of a 10 M_J planet at ≈0.6 au. Ground-based relative astrometry by Gatewood & de Jonge (1995) also rules out >10 M_J companions with periods of 1.2–5 yr, based on five years of relative astrometric observations of Altair, with a precision of ≈0.9 mas.

Altair is a challenging target for precise radial velocity studies, because it has few absorption lines due to its early type, and its lines are broadened due to fast rotation (Howard & Fulton 2016) and Stark broadening. The systemic velocity of Altair is constrained with high-resolution spectroscopy to within an uncertainty of 3.5 km s⁻¹ (Prieto et al. 2004). With repeated sampling, this would be enough to detect the 18 km s⁻¹ semiamplitude of a "dark" solar-mass companion in an edge-on orbit at 1 au, but the inclination of any companions to Altair remains unknown.

Direct imaging can help fill a niche here, even though Altair may be relatively old. It has isochrone-based age estimates of 0.7 Gyr (Vican 2012) to 1.2–1.4 Gyr (de Souza et al. 2005), though some evidence from hydrogen mass fractions suggests that Altair has just left the zero-age main sequence, and may be as young as ∼100 Myr (Peterson et al. 2006; Bouchaud et al. 2020). In any event, Altair is close enough that planets on orbits of 1 au would go out to a maximum of 02, or 3.4λ/D in the K band and 2.0λ/D in the $L^{\prime}$ band of an 8 m adaptive optics (AO)-equipped telescope. Since Altair is an early-type star, planets would also re-emit thermally at wider orbits than around cooler stars. In addition, in thermal equilibrium the level of re-emission is less affected by the system's age. It can be expected that thermal re-emission alone will cause a system with an "old" Jovian planet on a 2 or 3 au orbit around an A-type star to have a planet-to-star contrast of ∼10⁻⁸ or 10⁻⁹ in the $L^{\prime}$ band.

For these reasons, Altair warrants close investigation in its own right with direct imaging in the thermal infrared, as a complement to imaging surveys that put constraints on gas giant planets around young stars on wide orbits of tens of au (e.g., Males et al. 2014; Quanz et al. 2015; Meyer et al. 2018; Wang et al. 2019). Currently, however, no companions of any mass are known to orbit Altair, even though the star has been included in a number of direct imaging surveys (Kuchner & Brown 2000; Schroeder et al. 2000; Oppenheimer et al. 2001; Leconte et al. 2010; Roberts 2011; Janson et al. 2011; Dieterich et al. 2012; Stone et al. 2018). The current tightest constraints on companions around Altair from direct imaging have essentially ruled out ≳10 M_J planets for orbits with projected radii of ≳10 au, and ≳5 M_J planets at ≳20 au, for a stellar age set to 500 Myr (Janson et al. 2011).

In the meantime, other aspects of Altair have been characterized and have set the stage for detailed study of whatever exoplanet system it may host. Near-infrared and optical long-baseline interferometry (OLBI) have revealed the star's angular width, high inclination, asymmetric surface brightness, oblate spheroid shape due to its spin, and variable K-band-emissive exozodiacal dust in tight orbits of ∼0.1 au, though no emission has been detected from a cold debris disk or N-band-emissive habitable-zone dust (Kuchner et al. 1998; Ohishi et al. 2004; Reiners & Royer 2004; de Souza et al. 2005; Suarez et al. 2005; Peterson et al. 2006; Monnier et al. 2007; Richichi et al. 2009; Lara & Rieutord 2011; Millan-Gabet et al. 2011; van Belle 2012; Absil et al. 2013; Gáspár et al. 2013; Hadjara et al. 2014; Mennesson et al. 2014; Thureau et al. 2014; van Lieshout et al. 2014; Baines et al. 2017; Kirchschlager et al. 2017; Nuñez et al. 2017; Ertel et al. 2018, 2020; Bouchaud et al. 2020). van Belle et al. (2001) also note that visibility data from the Palomar Testbed Interferometer (PTI) consistent with an elliptical shape for Altair are probably not an artifact of a tight binary companion, because Altair's proper motion and the sensitivity of the PTI do not bring apparent binary companions within a range that could cause them to masquerade as an elliptical stellar surface.

Further constraints on the presence of companions to Altair will inform efficient survey design of space-based missions, for which observing time will be at a particular premium. Indeed, Altair is on the target lists of the Exo-C, Exo-S, and Nancy Grace Roman Space Telescope ⁹ study teams (Howard & Fulton 2016), and is among the best candidates for detecting and characterizing planets with a starshade coupled to Roman (Romero-Wolf et al. 2021). Simulations suggest that Altair is also one of the best candidates for finding Jovian companions from the ground in the thermal infrared, with the METIS instrument of the future 39 m European Extremely Large Telescope (ELT; Bowens et al. 2021).

The Large Binocular Telescope (LBT), on Mt. Graham in southern Arizona, is in a unique position for obtaining high-angular-resolution, thermal infrared imaging akin to that of ELTs. The LBT has twin 8.4 m telescopes equipped with AO (e.g., Hill 2010; Rothberg et al. 2018), with a 14.4 m center-to-center separation that provides a maximum 22.65 m baseline (Figure 1). The beams from the two telescopes can be coherently combined, in a "Fizeau" mode that preserves the coherence envelope across the field of view.

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In this work, we carried out high-contrast imaging of the star Altair in the LBT Fizeau mode. Our work represents the first filled-aperture LBT Fizeau data set with some degree of active phase control with a correcting mirror to counteract the time-dependent differential piston between the two telescopes and freeze the Fizeau fringe pattern. We developed a new code base for reducing these unique data, using a principal component analysis (PCA)-based decomposition to tolerate greater PSF phase diversity. In this analysis we increase the typical proportion of frames used from 2%–20% from the handful of past LBT Fizeau observations for static image deconvolution (Leisenring et al. 2014; Conrad et al. 2015; Conrad 2016) to 57%, and we place constraints on companions to Altair closer in to the star than in any previously published direct imaging data set. (Works by de Kleer et al. (2017, 2021) selected frames based on rapid-cadence changes in the observed object.)

In Section 2.2 we describe the observations, and in Section 2.3 we describe the unique systematics of the PSF and the reduction process. In Section 3 we discuss the results in the angular regimes of both classical direct imaging and Fizeau imaging. We discuss the results in Section 4, describe strategies for future improvement in Section 5, and conclude in Section 6.

2.1. LBT in Fizeau Mode

The LBT interferometer (LBTI) receives the telescope beams from the LBT tertiary mirrors, feeds <1 μm light to AO wave front sensors, and sends >1 μm light into the Nulling and Imaging Camera (NIC) cryostat (Hinz et al. 2008, 2016; Bailey et al. 2014). There, each beam branches into a shorter-wavelength beam to the phase camera (Phasecam; Defrère et al. 2014) and a longer-wavelength beam to the 1.2–5 μm LMIRCam (Skrutskie et al. 2010; Leisenring et al. 2012) and/or the 8–12 μm Nulling Optimized Mid-Infrared Camera (NOMIC; Hoffmann et al. 2014). (See Figure 2 for a schematic.)

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To combine the twin beams coherently in a Fizeau configuration, the entrance pupil (the twin primary mirrors) and the exit pupil (a stop internal to the instrument) share the same spatial proportions, which effectively makes the twin telescopes act as one large telescope aperture with a mask that passes two circular subapertures. This configuration allows the center of the coherence envelope to exist across the field of view. The illumination on the detector is still a simple convolution of the object with the telescope PSF, but the PSF is now a multiplication of an Airy function with a corrugation pattern due to the separation of the telescopes.

The Fizeau PSF samples Fourier space at frequencies higher than from a single filled aperture. Figure 3 shows the theoretical resolving power of the LBT based on the modulation transfer function (MTF), or the amplitude of the Fourier transform of the LBT PSF. The Fizeau PSF is also predicted to have contrast gains when searching for low-mass companions in the additional dark regions of the PSF, if detector integration times are shorter than the atmospheric coherence timescale τ₀ (Patru et al. 2017a).

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2.2. Observations

Our observations were carried out with the LBT in Fizeau mode on UT 2018 May 7, with conditions as tabulated in Table 1. We followed our standard procedure for setting up for Fizeau observations. After closing both AO loops, a grism was inserted into the science beam. The two grism illuminations (one from each telescope) were manually and incoherently overlapped on the LMIRcam science detector by moving tip–tilt mirrors internal to the instrument beam combiner (Hinz et al. 2004; Leisenring et al. 2012).

Table 1. Observing Log, UT 2018 May 7

Parameter	Value
Epoch (hh:mm UT)	10:15–12:16
Seeing (line-of-sight)	Mostly 09–13
Precipitable water vapor a	Mostly 8–13 mm H2O
Parallactic angle q	−43° ≤ q ≤ +3°
Airmass	1.1 to 1.2
Wind	0.5 to 4.5 m s−1
AO frequency b	1 kHz
AO deformation modes	300
AO control radius r c	4.05 μm/(2d) ≈ 14

Notes.

^a Measured at the Heinrich Hertz Submillimeter Telescope. ^b Listed AO parameters are for both telescopes. ^c d: subaperture spacing.

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An internal mirror was manually moved in piston to seek the coherence envelope of the two overlapping grism PSFs. This coherent combination of the beams leads to a "barber pole" or "candy cane" fringing pattern along the wavelength-dependent length of the combined grism illumination. Vertical fringes indicated an optical path difference of zero. The grism was then removed, leaving a Fizeau–Airy PSF (that is, a Fizeau PSF that is not chromatically dispersed) on the science detector. Ks-band fringes from a separate channel were placed on the phase detector by moving a beam combiner optic, and then the phase loop was closed (e.g., Spalding et al. 2019).

The closed phase loop acts to make another mirror upstream of the science and phase detectors move in piston, tip, and tilt at a rate of hundreds of hertz to 1 kHz. Movements along each of these three degrees of freedom are centered around tip–tilt and piston setpoints, which can be set manually to remove residual aberrations from the PSF.

Given the target brightness, we observed through a narrowband Brackett-α filter (4.01 ≤ λ/μm ≤ 4.09 at cryogenic temperature) and a 10% neutral density (ND) transmission filter, so as to saturate only the central bright Fizeau fringe peaks. Our science filter is similar to that of the NACO NB4.05 filter (Rodrigo et al. 2012; Figure 4). Unsaturated PSF frames were taken by substituting the 10% transmission filter with a 1% transmission filter. LMIRcam detector readouts of 2048 × 512 pixels, or a quarter of the array, were read out with an ISDEC controller backend (Burse et al. 2016). All frames had integration times of 0.146 s, with one detector reset for each read.

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Four different permutations of filters were combined in series upstream of the science detector. We denote the sets of detector readouts taken with each filter combination as "blocks" of frames, all four of which are listed in Table 2. Two of those blocks (A and D) were the saturated science frames. The other two (blocks B and C) were the unsaturated frames for reconstructing the PSFs. In blocks C and D, we added a wideband 2.8–4.3 μm filter. This filter, frequently used for spectroscopy, removes some of the strong water vapor emissivity in the M band.

Table 2. Blocks of Frames by Filter Combination

Frames	Filters (in Series)	Parallactic Angle q
"A" block (saturated science frames)	ND 10% transm., Brα (4.05 μm)	−433 to −360
"B" block (unsaturated)	ND 1% transm., Brα (4.05 μm)	−360 to −331
"C" block (unsaturated)	ND 1% transm., Wide 2.8–4.3 μm Brα (4.05 μm)	−321 to −288
"D" block (saturated science frames)	ND 10% transm., Wide 2.8–4.3 μm Brα (4.05 μm)	−220 to +33
Total Δq, saturated science frames	⋯	326°

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In the stochastic seeing conditions, the phase loop repeatedly opened and had to be manually reclosed. The longest duration for which the phase loop was continuously closed was 4.4 minutes. In addition, there were occasional fringe "jumps" on Phasecam by $1{\lambda }_{{K}_{s}}$ wavelength, which caused a jump of 0.53λ_Brα on LMIRCam (Maier et al. 2018, 2020; see also Table 2 in Spalding et al. 2018). The telescopes were nodded only once during the entire observation, so as to keep the PSF stable and reduce overheads.

No new companions to Altair were found in the data, though the analysis described in Section 2.3 characterizes the sensitivity of this novel mode of observation. Figure 5 shows an example of a reduced, classical angular differential imaging (ADI) image with an injected companion. The region of the readout used in the ADI reduction was restricted to a circle for computational expediency, and the central region of the star was masked after the subtraction of the stellar PSF. Since the PSF also contained Fizeau fringes, an additional analysis was performed for the angular regime of Fizeau fringes.

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2.3. Analysis

Traditional host star subtraction routines based on ADI make use of the fact that a faint astrophysical signal near the host star will rotate with the sky. The PSF is fixed relative to the pupil while the sky (and any potential companion) rotates beneath it, which allows decorrelation of the astrophysical signal from systematic effects such as quasistatic speckles.

Code bases have already been developed to reduce AO-corrected direct imaging data to search for low-mass companions. However, these routines have tended to be applied to stable, AO-corrected Airy PSFs, which are, to first order, axisymmetric. By contrast, the LBT Fizeau PSF is non-axisymmetric and exhibits degrees of freedom that are unique to the coherent combination of two Airy PSFs. These include differential tip, tilt, and phase (Spalding et al. 2019), all three of which can vary with time.

To deal with this unique and time-variable PSF, we constructed a new reduction pipeline that treats each detector readout individually, under the assumption that the morphology of the PSF can change significantly from frame to frame. This pipeline also treats large-angle, classical ADI regimes (ρ ≳ λ/D) and small-angle Fizeau regimes (λ/D ≳ ρ ≳ λ/B_EE) separately. We describe the pipeline in the following sections, with more details in Appendix A.

2.3.1. Regime of Classical ADI Imaging: ρ > λ/D

This part of the analysis is similar to that of classical direct imaging, with an AO-corrected PSF of size scale λ/D = 0101. Modeled values of the atmospheric coherence timescale at the top of Mt. Graham can be ≈1.5–2 ms toward the zenith at 0.5 μm ¹⁰ (see Turchi et al. 2016). For 4 μm observations at zenith angles γ of 25°–35°, the scaling law τ₀ ∝ λ^6/5(cos γ)^3/5 (e.g., Roddier 1999) yields coherence timescales of ≈20 ms. With no phase control, the integration time of 146 ms will lead to smearing of the fringes. Thus, out of saturated frames representing 12.9 minutes of integration, we selected only frames that had phase control. The closed phase loop brought the phase rms values as low as ≈0.3 μm. This satisfies the condition of piston rms ≪ λ, which maximizes fringe contrast in a given frame (Patru et al. 2017b), barring phase jumps. The science data that passed these quality checks represented a total of 7.4 minutes of integration (Table 3).

Table 3. Subsets of Science Frames Used for ADI

Frames	Relevant Regime	Parallactic Angle q	Angle E of N of Long Baselines a	Integration Time b tint	Efficiency c
Blocks A and D	λ/D	−433 to −360, −220 to +33	—	445 s d	0.093/0.57
Block A, strip 0	λ/BEE	−433 to −360	5032 (0E) and 23032 (0W)	124 s	0.072/0.45
Block D, strip 1	λ/BEE	−215 to −157	70782 (1E) and 250782 (1W)	103 s	0.060/0.82
Block D, strip 2	λ/BEE	−157 to −94	7657 (2E) and 25657 (2W)	75 s	0.098/0.60
Block D, strip 3	λ/BEE	−93 to −30	8337 (3E) and 26337 (3W)	82 s	0.111/0.68
Block D, strip 4	λ/BEE	−30 to +33	9004 (4E) and 27004 (4W)	61 s	0.084/0.52

Notes.

^a The naming convention of the baselines is a chronological number with a letter to indicate whether a half-baseline is the eastern or western half (see Figure 7). ^b The total integration time of frames in this subset, which were ultimately used in the reduction. ^c The first decimal is the integration time t_int used divided by wall-time elapsed during that section(s) of the observing sequence. This ratio is heavily affected by science data transfer and management overheads. The second number is the ratio of science frames used to all the science frames (or t_int out of the total integration time). The lost efficiency of this number is primarily due to phase loop openings. The cumulative frames used in the λ/B_EE reductions are the same as those in the λ/D reduction, so the total percentage of frames used in the λ/B_EE reductions is also 57%. ^d Note that the borders of ≈70% of the science frame cutouts centered on the PSF extend beyond the edge of the readout region by 1''. This brings the effective integration time of the science frames down to ≈130 s for a region >1'' to the southeast of Altair.

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Each host star PSF in a frame was reconstructed twice: once with a PSF PCA basis S set made from saturated frames (i.e., those in blocks A and D), and once with a basis set U made from unsaturated frames (those in blocks B and C). Unsaturated reconstructions of saturated frames were made by projecting the science frames onto U, with saturated pixels in the innermost bright fringes reconstructed based on the projection of the rest of the PSF on the basis set.

The reconstruction of the unsaturated PSF for a given frame was injected into the frame as the fake planet, whose PSF shares the host star's Fizeau aberrations of differential tip/tilt and phase. (See Figure 6.) In this way we minimize the introduction of systematics stemming from the forcing of the fake planet PSF to fit a fixed PSF model, such as a median across a stack of frames.

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Fake companions were injected at ϕ = {0°, 120°, 240°} east of true north, and at distances from the host star corresponding to ρ = FWHM × {2, 3, 4, 5, 7, 9, 12, 15, 19}, where FWHM = 0104 is the FWHM of a perfect Airy PSF. An entire reduction was done for every single combination of (ϕ, ρ), one fake companion in a data set at a time. The inner cutoff of 2FWHM is somewhat arbitrary, but was made to allow overlap with the treatment of the λ/B_EE regime (Section 2.3.2).

After the host star was subtracted from each frame with the reconstruction based on a projection onto the basis set S, the images were derotated and a 2D median was taken across the frames to produce a single ADI frame. This frame was convolved with a 2D Gaussian of size λ/D to remove the effects of pixel-to-pixel noise. The signal was taken to be the maximum value of counts within a circular cutout centered on the fake companion, with a width of one FWHM of the Airy function. The noise was calculated as the standard deviation of counts in a series of "necklace-bead" patches along the same annulus as that containing the fake companion. The centers of the patches are circumferentially separated from each other by one FWHM.

The pipeline repeated the reductions iteratively, perturbing the fake planet amplitudes until convergence to signal-to-noise ratio (S/N) = 5 at a given location (ϕ, ρ). An azimuthal median of the companion amplitudes was made to generate a 1D "classical" contrast curve, which was then modified to account for diminishing degrees of freedom at small radii, following the framework of Mawet et al. (2014). This framework also allows a false positive fraction that varies as a function of radius and a true positive fraction that is fixed at 0.95. More details of this part of the reduction are described in Appendix A.2, and Appendix C provides more details on the calculation of the contrast curves.

2.3.2. Regime of Fizeau Fringes: λ/D ≳ ρ ≳ λ/B_EE

At 4.05 μm, the Fizeau regime can in principle carry information down to λ/B_EE = 0037. However, the Fizeau fringes effectively rotate around the object during the observation, because the fringes are stationary on the detector while the sky rotates. ADI applied to all the frames at once would then only have the effect of washing out the fringes. We therefore subdivided the science frames into subsets listed in the rows corresponding to λ/B_EE in Table 3. It should be noted from Table 3 that preserving the Fizeau fringes by subdividing the data set comes at the cost of total integration time for each subset of frames.

To subtract the host star, we use PCA-reconstructed regions along rectangular strips. In one set of reductions, a strip was set with the long axis along the long (Fizeau) LBT baseline. In another set of reductions, the reconstructed region is along the short LBT baseline, which would most closely approximate classical imaging. Those regions are overlapped for illustration in Figure 7 to show how they rotate on the sky. These perpendicular orientations were chosen to compare the results based on the longest and shortest LBT baselines, though baselines could be chosen for any arbitrary orientation.

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To determine sensitivity to a companion along a given baseline, frames are reduced with injected fake companions ranging across a grid of amplitudes equivalent to the star-to-planet contrast 0 < Δm < 10, and separations equivalent to 0.28 < ρ/(λ/B_EE) < 14.1. Again, reductions are always performed with one fake companion at a time. This is done for all frames, including those for strips that have angles different from the baseline whose sensitivity we want to determine. This allows for a companion PSF that has a center outside a given strip to "bleed" into the other strips on the other baselines, as a true PSF would.

After subtraction of the PCA-reconstructed host star, a series of two-sample Kolmogorov–Smirnov (KS) tests is done with pairs of 1D residuals of strips. We use the two-sample KS test to determine the acceptance or rejection of the null hypothesis H₀, which states that two empirical distributions come from the same parent sample, with a confidence of 95%. This is effectively a method of detecting azimuthal anomalies that betray the presence of a companion as the long baseline of the LBT sweeps out the area around the host star (Figure 8).

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The residuals of the strip with planets injected along it are compared with the residuals of other strips pointed in the nearest cardinal direction. These KS grids in (ρ, Δm)-space are then averaged into one grid corresponding to the closest cardinal direction N, S, E, or W. The critical value of the contour plot of the KS landscape is that which divides detections with 95% confidence and nondetections. Appendix A.3 provides more details of this stage of the pipeline and shows that the critical dividing line corresponds to the contour value 0.2716.

This section presents the results of the data analysis, in terms of contrast and sensitivity. We show a 1D contrast curve for the λ/D regime in Figure 9. The contrast curve was converted into absolute magnitudes (see Appendix B) and then to mass amplitudes with the use of four different models derived from the PHOENIX radiative transfer code (Baron et al. 2010) for standalone, self-luminous objects. They include AMES-Cond (Allard et al. 2001; Baraffe et al. 2003); BT-Cond, which incorporates updated molecular opacities (Plez 1998; Ferguson et al. 2005; Barber et al. 2006); and BT-Dusty and BT-Settl, which allow for suspended particulates (Allard et al. 2012).

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Figure 10 shows the mass sensitivity, which is limited to objects ≳0.5 M_⊙, well above the hydrogen-burning limit and the brown dwarf mass range of 13 ≲ M/M_J ≲ 80 (Spiegel et al. 2011). At these sensitivities the differences in the models between ages 1.0 and 0.7 Gyr were negligible, and so we consider only 1.0 Gyr in what follows.

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For the λ/B_EE regime, Figure 11 shows the landscape of the KS statistic. From these results, we cannot reject the null hypothesis for companions at radii smaller than ≈0.15'', or within 1 au. Figure 12 shows an example detection from along the critical KS contour at Δm = 7. If, however, a fake planet amplitude is comparable to that of the host star at small angles, the changing mode pattern used to subtract the host star exacerbates the residuals at those angles. This leads to the sensitivity inversion seen at very bright fake companions (0 < Δm < 2) in Figures 11 and 13.

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Figure 14 shows that in the regime of λ/B_EE the constraints based on fake companion injections and the KS test have sensitivities at the smallest angles corresponding to the highest masses that can be interpolated from the model grids, ≈1.3 M_⊙.

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We now contextualize our results within related work. Stone et al. (2018) conducted a filled-aperture direct imaging survey of 98 B- to M-type stars using either or both of the 8.4 m telescopes of the LBT. The LEECH observations did not involve coherent combination of the beams: when the two telescopes were being used simultaneously, the PSFs were physically separate on the LMIRCam detector. Altair itself was observed only using the left-side LBT sub-telescope, and the LEECH contrast curve for that target is shown in Figure 9.

This Fizeau data set does not outperform constraints from LEECH around Altair in the λ/D regime, but it also only has ≈15% of the integration time in the innermost arcsecond. One source of lost efficiency is the time overhead from each of the detector readouts. Another source is the highly time-dependent nature of the optical aberrations in the Fizeau PSF, which is mostly due to repeated openings of the phase loop and the need to manually realign and reclose the loop.

Figure 9 shows comparisons of this Fizeau λ/D contrast curve with that of the LEECH survey, based on simple scalings of the observation setup. Figure 9 includes the effects of rescaling the collecting area, allowing more rotation of the target on the detector, removing the neutral density filter, and breaking up the observation into separate epochs to decorrelate the noise. If these modifications are combined and speckles are sufficiently uncorrelated (Marois et al. 2006; Males et al. 2021), it would appear that the Fizeau mode can be competitive with single-aperture imaging at small angles.

In the λ/B_EE regime analysis, constraints on companions to Altair are as close in as ≈015. This inner working angle is closer in than the published constraints specific to Altair in any of the direct, non-interferometric imaging surveys mentioned in Section 1, which do not go closer in than 04 (Kuchner & Brown 2000; Leconte et al. 2010; Janson et al. 2011; Dieterich et al. 2012), and the unpublished Altair-specific curve in the analysis of Stone et al. (2018) goes as close in as 028, although the mass ranges of interest in these surveys were substellar. Among OLBI observations of Altair mentioned in Section 1, there has been no convincing evidence of a visibility profile perturbed by a very-close-in companion point source. (The X-ray emission of Altair was once thought to suggest chromospheric activity of a companion, but the emission has been found to be consistent with Altair's own corona (Robrade & Schmitt 2009).)

The most recent constraints on Altair's position angle and inclination are PA = 3011 ± 03 and i = 507 ± 12 (Bouchaud et al. 2020). If planets in the Altair system orbit in the plane perpendicular to Altair's spin axis, then the contrast curves in this work sample all of the projected habitable zone, which lies at 03 ≤ ρ ≤ 10. Figure 15 shows a projection of this zone, with lines overplotted to show the sampled long LBT baselines.

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It should be noted that our innermost constraints are set by the KS test applied to data with fake injected planets. In this context the KS test is weaker than fake planet recovery in the λ/D regime, because it effectively asks "are these strips along a given baseline drawn from different distributions?" and assumes that variations between strips at different angles will be introduced by the PSFs of companions. The subsets of frames in the λ/B_EE analysis each represent less integration time than that available for the λ/D reduction, by virtue of the fact that only small amounts of rotation can be tolerated to preserve the narrow Fizeau fringes. Averages of the contrast curves across subsets, which have a cumulative integration time that is the same as the data in the λ/D regime, are shown in Figure 13. Though these curves are still in the stellar mass regime, we place the closest-in direct imaging constraints on companions to Altair, at a sensitivity of 1.3 M_⊙, at ≈015. In this analysis we have also used the highest percentage of science frames from an LBT Fizeau data set, 57%, in both the λ/D and λ/B_EE regimes.

The nature of the Fizeau PSF, and its stability during these observations, suggests a variety of strategies for future improvement.

We chose integration times and filter combinations to avoid hard saturation of the entire Airy core and the structure of Fizeau fringes contained therein. However, this leaves low S/N in the Fizeau PSF at radii corresponding to the λ/D regime, which could yet prove valuable for putting constraints on low-mass companions, particularly in the darkest regions of the Fizeau PSF (Patru et al. 2017a).

If the purpose of the observation is to examine both the λ/D and λ/B_EE regimes, we suggest taking future observations in a bifurcated way: a sequence of frames with hard saturation of the Airy core and high S/N in the larger surrounding Fizeau PSF structure; and a sequence of frames with only slight saturation of the innermost bright Fizeau fringes, to retain good visibility of the innermost dark Fizeau fringes. (In addition, entirely unsaturated frames should be taken to reconstruct the core of the host star PSF, as we have done in this work.) It may also be advisable to have a greater separation of the two regimes during the reduction process. For example, separate PCA basis sets could be generated for different tesselation patterns, and may minimize effects such as the inversion seen in Figure 13.

It is always important to consider weather conditions, but it is particularly important for delicate observations such as these. To increase the S/N in the λ/D regime, and to access angles in the λ/B_EE regime, clearly one of the most critical aspects is the stability of the phase loop—and that requires good seeing. Poor seeing scrambles the coherence of interfered light, which degrades the visibility of the fringes on the phase detector to below the level required by the phase software loop. How "good," then, does good seeing have to be to keep the phase loop closed?

Thus far, the great majority of LBTI observations taken with a closed phase loop have been for the HOSTS survey of exozodiacal dust disks (Ertel et al. 2020). In Figure 16 we compare seeing values during HOSTS observations ¹¹ with those taken in this work. HOSTS observations were either scheduled classically, or scheduled in queue mode when seeing was generally below 12. Regardless of how a stretch of closed-loop HOSTS observations began, the stochastic nature of weather conditions on Mt. Graham conspired to add considerable diversity in the seeing values. No closed-loop HOSTS sequences were successful with seeing of ≳15, and this should be considered a prohibitive level of turbulence for phase control with the current instrument configuration.

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However, Figure 16 also shows that the majority of our Altair observation was not during prohibitively poor seeing, and only indicates a weak dependence of the phase loop on the seeing value alone. Some other source of instability seemed to be present that forced the phase loop to open. In these observations of Altair, the seeing zigzagged with a typical amplitude of 01–02 on a timescale of 1–3 minutes. Sometimes the phase loop opened at sudden changes in the seeing—at jumps and drops (see Figure 17). The phase loop openings coinciding with shifts in the trend of the seeing might mark the passage of boundaries between large pockets of turbulence. It may be advantageous to restrict phase-controlled observations to periods in which seeing is smooth as a function of time, in addition to being low. The astronomer cannot control the seeing, but the Italian LBT partner Istituto Nazionale di Astrofisica (INAF) is developing the online Advanced LBT Turbulence and Atmosphere (ALTA) Center, ¹² which features a tool for predicting the nightly temporal evolution of various LBT-specific environmental parameters including the seeing (Masciadri et al. 2019). The delicacy of our phase loop closure demonstrates the importance of a reliable predictive tool for the summit of Mt. Graham, and we encourage the use of such a tool in the future.

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Since the LBT is an Alt-Az telescope with sub-telescopes always perpendicular to the elevation mount, the LBT cannot freeze the angle of the long LBT baseline relative to an arbitrary object for long integrations. An observer planning to sample certain baselines around an object will have to consider the time-dependent parallactic angle, and the heaviest sampling will occur when the angular change is slow. Accessible elevations are bounded by a hard lower telescope elevation limit of 30°, and a higher limit of ≳80° due to AO tip–tilt instability as the mount swings around rapidly during the meridian transit of the object (A. Vaz 2021, personal communication).

The analysis by Patru et al. (2017a) offers important guidance for considering how to sample the object as it rotates, given the morphology of the LBT Fizeau PSF. For example, along the long LBT baseline is the highest-frequency information, but for a given parallactic angle rotation, planet PSFs will drift through more of the dark host star fringes if they lie in the direction along the short baseline. Observing programs could be designed such that the LBT could move from target to target and back, so as to take snapshots of the long baseline at substantially different angles over multiple objects. For a given snapshot, the closest-in Fizeau fringes would offer the best contrasts at those angles, but the greatest amount of "drift" of a companion through dark fringes would happen if it moves perpendicular to them, at a wider angle from the star.

There are also mechanical considerations or upgrades to be made to LBTI. During the observations in this work, we nodded the Fizeau PSF on the detector a single time to avoid incurring any more PSF instability or manual realignments. After this data set was taken, sets of lenses were installed at staggered radii in a filter wheel upstream of the Phasecam detector. This could facilitate rapid side-to-side nodding in Fizeau mode by shifting the illumination on Phasecam with a wheel movement, and then nodding the telescope mount to shift the PSF on the detector and recovering the co-alignment. This could enable more dithering on the detector. Some on-sky tests of this technique have been performed, but it remains to be fully commissioned.

Phasecam itself was originally designed to target the bright stars of the HOSTS survey, and the detector readouts have noise at levels that can be prohibitive for dim targets. Furthermore, it had to read out at a rate that was pegged to the cadence of corrections from the telescope's vibration-monitoring system (Böhm et al. 2016) at ≥500 Hz, and the only "ground truth" measurement of changes in pathlength was the illumination on the Phasecam detector. The absence of any independent measurement of mirror movements between the snapshots of the phase added phase noise, in addition to the phase noise already inflicted by the atmosphere and telescope vibrations. These factors conspire to limit phase control to bright stars of m_K ≲ 4.7. However, there are plans to upgrade Phasecam in the near future with a SAPHIRA array (Goebel et al. 2018) with sub-electron read noise, to install capacitive sensors behind the pathlength corrector mirrors to measure mirror movements, and to develop the software to enable decoupling of the cadences of Phasecam and the telescope vibration-monitoring system. These upgrades are predicted to enable the phase loop to close on stars m_K < 9 (J. Stone 2021, personal communication). It remains to be seen how an upgraded phase-sensing detector will perform in unstable seeing as in Figure 17, but the target list for phase-controlled Fizeau observations will increase dramatically, and will include dimmer targets for which other improvements described in this section will be important.

The science camera LMIRcam has already been upgraded in the time since the observations taken for this work. The detector remains the same, but the ISDEC controller has been replaced with a MACIE interface card (Loose et al. 2018). By enabling a switch from USB2 to USB3, this has led to an order-of-magnitude increase in the data transfer speed, to 4.8 Gbps. The new MACIE also has a programming interface that allows more readout mode flexibility.

The MACIE electronics enables both of the native pixel readout rates of the LMIRcam detector: "slow" (100–500 kHz) and "fast" (5 MHz). Trade-offs between these modes include frame time, read noise, dark current, analog-to-digital convertor bit depth, and available subarray regions. The USB3 connection allows image data to be streamed continuously to the host computer's memory buffer and to be saved to disk without any data buffering overheads, significantly increasing on-source efficiency over the previous implementation. In addition, multiple nondestructive read frames can be acquired "up-the-ramp" for each integration to allow slope fits and reduce read noise. This also enables a substantial increase in the proportion of time spent integrating on a target relative to the total time the detector is active. In the readout mode at the 5 MHz pixel rate, for example, frame times are 28 ms/frame, and integrations and readouts of the full 2048 × 2048 array, with 10 nondestructive readouts per frame, would have an integration time efficiency of above 90%. Even after slope fitting effectively brings this down to ≈80%, this is much more favorable than the ≈10% efficiency in this work (see Table 3), which only used one-quarter of the readout array.

It may be even more economical to use subarrays that are smaller than the full detector, though a sufficiently large footprint is required around the host star to determine the background level around bright stars (see Appendix A.1). As a general guideline, we encourage observers to generate and reduce simulated data sets as they design their observing strategy, so as to make quantitative studies of the trade-offs of various LMIRCam or NOMIC readout modes for their particular science program, and to make maximum use of precious observing time.

On the software side, the phase loop can also be improved to increase the number of frames with good phase control. Phasecam currently uses only Ks-band light for the phase correction. As mentioned in Section 2.2, there are occasional phase jumps where the unwrapped phase skips an integer multiple of a Ks wavelength in pathlength, and introduces a phase error on the Fizeau PSF on the science detector. This happens frequently in unstable seeing. Until now, phase jump corrections have been made manually by the observers, but efforts are underway to supplement the phase loop with H-band light to swiftly and automatically correct these jumps (Maier et al. 2018, 2020).

Work has also been performed to develop a "correction loop" to supplement the phase loop (Spalding et al. 2018, 2019). The correction loop would automate alignments and use cutouts of the science detector readouts in real time to remove non-common path aberrations (NCPA) between the phase and science detectors. Some initial on-sky tests of alignment algorithms have been performed, but again, this loop remains to be fully commissioned. In this work we have tried to account for phase and other aberrations in the science PSF by PCA-decomposing each frame, but sensitive future observations in the λ/B_EE regime will require the Fizeau fringes to remain as frozen in place as possible, for as many frames as possible. Development of a correction loop to remove NCPA should continue.

Complementary to improvements of the phase control would be a comparative study of postprocessing techniques that could cope with loosened observing constraints. For example, in the λ/D regime, phase closure may be an overly restrictive requirement in the reduction. The phase jitter of the Fizeau fringes will add some noise to the background, and the study by Patru et al. (2017b) of the effects on the Fizeau PSF structure and a number of merit functions is an important step. Given the spatial heterogeneity of the potential contrast gain in the Fizeau PSF pointed out by Patru et al. (2017a), what is needed next is fake planet recovery in synthetic data sets with very dense "seeding" of fake planets throughout the stellar PSF structure.

In this work we have performed high-contrast imaging with the LBT Fizeau PSF on the star Altair, one of the nearest early-type stars. The inclination of the star is low enough that our contrast curves sample all phases of circular orbits in the habitable zone normal to the stellar spin axis.

The data reduction was separated into two regimes: one at radii ∼ λ/D from the host star, as is traditionally done in high-contrast imaging with AO-corrected PSFs, and one at radii ∼ λ/B_EE, which is unique to the LBT.

Even with AO correction, the Fizeau PSF exhibits unique aberrations of differential tip, tilt, and phase. We carried out an analysis that accommodates those additional degrees of freedom and treats every detector readout individually. In this manner we have been able to use 57% of the science frames.

These observations have a sensitivity for 1.0 Gyr down to ≈0.5 M_⊙ at large radii, roughly at the K- to M-type transition. More significantly, we place constraints on companions of masses ≈1.3 M_⊙ at 015 along the long baseline of the LBT. Constraints are similar along the short baseline, with a slightly better performance on the long baseline at most angles ≲025.

Future users of the LBT Fizeau mode should take into account the strong trade-off in sensitivity between the two radial regimes: signal-to-noise ratio at radii ∼ λ/D will be maximized with hard saturation of the Airy core, but the innermost Fizeau fringes at radii ∼ λ/B_EE will be preserved with shorter integration times.

In either regime, the most obvious mechanical need is for the ability to close the phase loop with minimal interruptions or downtime, and thereby maximize the number of usable readouts. This would also allow integration times longer than the atmospheric coherence timescale. It is therefore important to take these observations in good and stable seeing.

When the PSF stability is good enough and a more exacting selection can be made of the data, further improvement may be sought by using the merit criteria of Patru et al. (2017b). When the sensitivity becomes competitive with pre-existing filled-aperture data sets, data reductions with fake planet PSFs dispersed in more locations around the host star will also allow testing of the contrast gain maps of Patru et al. (2017a).

We acknowledge Jordan Stone's assistance during the LBTI observations that went into this work. We also thank the dedicated LBT support personnel for all their assistance to make an operation like the LBT possible, and to Steve Ertel, Tomas Stolker, Kevin Wagner, and George Rieke for insightful discussions. The LBTI itself is funded by the National Aeronautics and Space Administration as part of its Exoplanet Exploration Program, and LMIRCam was funded by NSF grant 0705296.

This material is based upon High Performance Computing (HPC) resources supported by the University of Arizona TRIF, UITS, and Research, Innovation, and Impact (RII) and maintained by the UArizona Research Technologies department. We also gratefully acknowledge a special project allocation from the UA Research Computing HPC.

The material in this paper is also based upon work supported by CyVerse cyberinfrastructure, which is funded by the National Science Foundation under Award Numbers DBI-0735191, DBI-1265383, and DBI-1743442. URL:www.cyverse.org. We also thank CyVerse for support to attend the 2018 CyVerse Container Camp.

Thanks go to Blake Joyce and Julian Pistorius for their assistance, which was made possible through the University of Arizona Research Technologies Collaborative Support program, and Blake and Julian's support of PhTea and Hacky Hour.

This made use of the SIMBAD database, operated at CDS, Strasbourg, France (Wenger et al. 2000); and the SVO Filter Profile Service (http://svo2.cab.inta-csic.es/theory/fps/) and VOSA, developed under the Spanish Virtual Observatory project, supported by the Spanish MINECO through grant AYA2017-84089. VOSA has been partially updated by using funding from the European Union's Horizon 2020 Research and Innovation Programme, under Grant Agreement no 776403 (EXOPLANETS-A).

This publication also made use of data products from the Two Micron All Sky Survey (2MASS), which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.

The authors also wish to recognize and acknowledge the significant cultural role and reverence that the summit of Mt. Graham (in Apache Dzi ł Nchaa Si'an, "big seated mountain") holds within the indigenous White Mountain and San Carlos Apache communities.

Part of this work has been carried out within the framework of the National Centre of Competence in Research PlanetS supported by the Swiss National Science Foundation. S.P.Q. acknowledges the financial support of the SNSF.

K.M.'s work is supported by the NASA Exoplanets Research Program (XRP) by cooperative agreement NNX16AD44G.

E.S. was supported by the Ed and Jill Bessey Scholarship in Astrobiology, and is currently supported by Jeff Chilcote under NSF MRI grant 1920180.

Facility: LBT/LBTI. -

Software: Python (Van Rossum & Drake 1995; Oliphant 2007), matplotlib (Hunter 2007), numpy (Walt et al. 2011), astropy (Price-Whelan et al. 2018), scipy (Virtanen et al. 2020), Singularity (Kurtzer et al. 2017), Singularity Hub (Sochat et al. 2017), Binder (Project Jupyter et al. 2018).

Compared to an Airy PSF, the Fizeau PSF has additional degrees of freedom in its aberrations. The amplitudes of these aberrations were also highly time-variable during this observation, which required the host star PSF to be modeled on a frame-by-frame basis in the reduction pipeline, which we describe here. The functionality of the pipeline can be split into three main sections corresponding to those numbered in Figure 18.

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A.1. Section 1: Data Preparation

A.2. Section 2: λ/D Analysis

To begin the process of subtracting the host star PSF and injecting fake planets, a median science frame was subtracted from the stack of science frames. The host star was then decomposed twice: with PCA basis set S, corresponding to the residuals of a saturated PSF to enable best host star subtraction; and basis set U to generate fake planet PSFs. For the latter, the saturated regions of the science frame (defined as pixels where detector counts >55k) are masked during the decomposition, so as to reconstruct the full PSF within the saturated regions. (However, the saturation only involved the innermost bright lobes of the Fizeau PSF so as to leave the innermost dark Fizeau fringes unsaturated.) The full frames of the residuals are then projected onto the basis set U. The host star amplitude was determined by taking an ADI image of the reconstructed host star PSFs, convolving the image with a Gaussian kernel, then taking the maximum value. A fake planet generated from the reconstruction U was then injected at a given starting amplitude and radius and angle from true north (ρ, ϕ). It is worth emphasizing that as the host star PSF changes from one frame to the next, the shifting of the Fizeau fringes and other PSF distortions are stamped into the fake planet PSF for that frame, just as the aberrations between a host star and true planet PSF would be correlated.

All of the preceding describes the process of preparing frames and injecting a first round of fake planets for both the λ/D and λ/B_EE analyses, but it is at this point that the two analyses split into two strictly separate treatments. To carry out the PCA decompositions for subtracting the host star, each science frame for the λ/D analysis was split into a tesselation pattern, where regions were sized to be at least 10× the area π(FWHM/2)² ≈ 74 pixels², the area of the footprint of an Airy disk (Figure 19). This was done to optimize the subtraction of the host star PSF in individual regions, but without overfitting to the basis set (Lafrenière et al. 2007). Each region was independently projected onto the basis set S and the resulting reconstruction was subtracted from that region. The outside edges and the centermost circle do not correspond to regions for decomposition in the λ/D regime and are ignored. Following the subtraction of every region from each frame, the frames were derotated according to their parallactic angles, and a median was taken across them to yield a single ADI frame. (Since the LBT is an Alt-Az telescope and LBTI has no derotator, the PSF does not rotate with the sky. Thus the PSF decomposition must be done before derotating the frames.)

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For a final ADI frame, the signal-to-noise ratio of a fake planet was calculated as the maximum counts within an FWHM-sized aperture around the fake companion, divided by the empirical standard deviation of counts sampled from a string of "necklace-bead" patches along an annulus at the same radius as the fake companion. The centers of these patches were separated from each other by a circumferential distance of FWHM, and excluded the companion itself (see Mawet et al. 2014). Each noise patch was defined to contain the pixels within a radius of 0.75 pixel of the calculated center of the patch. The value of counts from the patch was taken to be the median value of counts among the pixels in that patch. The pipeline iterated the fake planet amplitudes and repeated reductions of each data set until the signal-to-noise ratio of a companion converged to 5.

A.3. Section 3: λ/B_EE Analysis

At radii corresponding to the innermost Fizeau fringes (λ/D ≳ ρ ≳ λ/B_EE), information is washed out if the PSFs are all derotated and "medianed." The degrees of freedom for placing a companion around the host star also diminish, to the point where radii from the star become small enough that the circumference at that radius is ∼1 FWHM.

To treat this regime, we injected fake planets in the frames as in Section A.2, but with a number of key differences. First, fake planets are injected corresponding to a preset grid of (amplitude, radius) space, without converging on an amplitude iteratively. Second, reductions are performed with subsets of frames that span narrow ranges of parallactic angle. Third, the tessellation of the science frame, which is used to project regions onto basis sets, consisted of a single rectangle with a long axis along the long LBTI baseline, and a second identical region along the short LBT baseline. (See Figure 19 and Section 2.3.2.) By design there was not much angular rotation in each reduction, so the residuals of the resulting ADI frame had a footprint similar to that of the tessellation region.

Fourth, we chose a different way of quantifying the residuals after host star subtraction. We generate 1D residuals by taking the median along the 7 pixel wide short axis of the tesselation region (the strip), to generate a 1D set of residuals corresponding to the long axis. The 7 pixel width was chosen to be wide enough so as to take a median over several pixels and produce a 1D set of residuals as a function of radius that does not exhibit pixel-to-pixel noise, but still narrow enough to be unaffected by the loss of effective integration time along the edges of the strip following ADI. Sets of 1D residuals are compared with each other as long as they are not immediately adjacent to each other (unless the strip of interest is 0E, which is separated from the other strips by a larger position angle). We do not compare half-strips pointed in different cardinal directions. This reduces the parallactic angle diversity among the compared strips, but avoids the misidentification of quasistatic asymmetries in the residuals around the Fizeau PSF as a "companion."

For example, the strips along the baseline pointing closest to east are denoted 0E, 1E, 2E, 3E, and 4E. If a fake planet was injected along the position angle of half-strip 3E, then comparisons are made between the 1D residuals of 3E and each of the half-strips 0E and 1E. The half-strips 2E and 4E are excluded, because they are immediately adjacent to 3E and contain a significant amount of the fake planet PSF centered in strip 3E so that it would be comparable to making a comparison between the same 1D residuals—which would act to decrease sensitivity to companions along strip 3E. If, on the other hand, a fake planet was injected along the position angle of half-strip 0E, then the residuals are compared with each of 1E, 2E, 3E, and 4E.

Finally, we use a different statistical test than in the λ/D regime for deciding whether a companion has been detected. For the λ/B_EE regime we used the Kolmogorov–Smirnov (KS) two-sample test to compare pairs of the 1D residuals. This KS two-sample test either accepts or rejects the null hypothesis H₀ that the residuals do not come from different parent distributions (Conover 1971). The KS statistic is the maximum difference between the two cumulative distribution functions (CDFs), or

$\begin{eqnarray}&&D=\max \left|{S}_{n}(x)-{S}_{m}(x)\right|\end{eqnarray} \tag{ A1 }$

where we use S_n (of length n) and S_m (of length m) to denote the two CDFs of the sets of empirical residuals.

The probability of this value being less than a critical value converges to the Kolmogorov–Smirnov cumulative distribution function L(z) as in Feller (1948),

$\begin{eqnarray}&&\begin{array}{c}\begin{array}{l}{\rm{Prob}}\left\{D\leqslant z\sqrt{\displaystyle \frac{m+n}{{mn}}}\right\}\longrightarrow L(z)\\ \,\equiv 1-2\displaystyle \sum _{\nu =1}^{\infty }{\left(-1\right)}^{\nu -1}{e}^{-{\nu }^{2}{z}^{2}}.\end{array}\end{array}\end{eqnarray} \tag{ A2 }$

We reject H₀—that the distributions are indeed from the same population—with a confidence of 95% at a value of z = 1.358 (e.g., Smirnov 1948).

We compare the residuals of all the half-strips with each other to generate KS statistic distributions across the grid of injected planet parameters (i.e., the grid of amplitude and radius from the host star). A median is taken across the distributions from all half-strip comparisons, and the critical value contour of the median landscape is considered to be the contrast curve. Beyond that contour, where $D\gt z\sqrt{(m+n)/{mn}}$ from a comparison of two strips (one of which has a planet injected along its median angle), the hypothesis H₀ is rejected. In this analysis, we compare samples of the values of pixels along radii of n = m = 50 pixels around the host star, so the critical value is ${D}_{{\rm{crit}}}=z\sqrt{(m+n)/{mn}}=1.358\sqrt{2/50}=0.2716.$

To find the magnitude of Altair at wavelengths relevant for this work, we write the apparent magnitude of a star on the Vega scale in terms of the mean flux and a zero-point:

$\begin{eqnarray}&&{m}_{\mathrm{star}}=-2.5{\mathrm{log}}_{10}(\langle {f}_{\lambda }{\rangle }_{P})+2.5{\mathrm{log}}_{10}({\mathrm{zp}}_{P}({f}_{\lambda ,\mathrm{Vega}}^{(0)}))\end{eqnarray} \tag{ B1 }$

where the P subscript makes it explicit that these quantities are to be calculated for a photon detector like LMIRcam, not an energy detector. The zero-point for an energy detector (subscript E) for the Paranal NACO NB405 filter is defined by the Virtual Observatory SED Analyzer (VOSA; Bayo et al. 2008) as

$\begin{eqnarray}&&{\mathrm{zp}}_{E}({f}_{\lambda ,\mathrm{Vega}}^{(0)})\equiv \displaystyle \frac{\int d\lambda {R}_{\lambda }(\lambda ){f}_{\lambda ,\mathrm{Vega}}^{(0)}(\lambda )}{\int d\lambda {R}_{\lambda }(\lambda )}\end{eqnarray} \tag{ B2 }$

with units of erg cm⁻² s⁻¹ Å⁻¹. We calculated this zero-point using the Vega spectrum from the Python package pysynphot, checked the result with the SVO value, and then calculated our own photon detector zero-point as

$\begin{eqnarray}&&{\mathrm{zp}}_{P}({f}_{\lambda ,\mathrm{Vega}}^{(0)})\equiv \displaystyle \frac{\int d\lambda {R}_{\lambda }\lambda {f}_{\lambda ,\mathrm{Vega}}^{(0)}}{\int d\lambda {R}_{\lambda }\lambda }\end{eqnarray} \tag{ B3 }$

where we have dropped the (λ) arguments to compactify the notation. The average photon flux of a star at the top of the Earth's atmosphere is the average flux at the surface of the star, scaled for distance:

$\begin{eqnarray}&&\langle {f}_{\lambda }{\rangle }_{P}=\displaystyle \frac{\int d\lambda {R}_{\lambda }\lambda {f}_{\lambda ,\mathrm{star}}^{(0)}}{\int d\lambda {R}_{\lambda }\lambda }={\left(\displaystyle \frac{{R}_{\mathrm{star}}}{D}\right)}^{2}\displaystyle \frac{\int d\lambda {R}_{\lambda }\lambda {f}_{\lambda ,\mathrm{star}}^{(\mathrm{surf})}}{\int d\lambda {R}_{\lambda }\lambda }.\end{eqnarray} \tag{ B4 }$

For the stellar spectrum, we used a Kurucz synthetic spectrum with specifications T_eff = 7750 K, log(g) = 4.0, and [Fe/H] = 0 (Castelli et al. 1997). This is to mimic Altair with T_eff = 7550 K, log(g) = 4.13, and [Fe/H] = −0.24 (Erspamer & North 2003), though the science wavelengths are well into the Rayleigh–Jeans regime for a star of this temperature.

Putting everything together, we find

$\begin{eqnarray}\begin{array}{c}\begin{array}{c}\begin{array}{rcl}{M}_{{\rm{star}}} & = & -2.5{{\rm{log}}}_{10}\left\{{\left(\displaystyle \frac{{R}_{{\rm{star}}}}{D}\right)}^{2}\displaystyle \frac{\displaystyle \int d\lambda {R}_{\lambda }\lambda {f}_{\lambda ,{\rm{star}}}^{\left({\rm{surf}}\right)}}{\displaystyle \int d\lambda {R}_{\lambda }\lambda {f}_{\lambda ,{\rm{Vega}}}^{\left(0\right)}}\right\}\\ & & -5{{\rm{log}}}_{10}\left(\displaystyle \frac{d}{10\,{\rm{pc}}}\right)=\mathrm{1.87.}\end{array}\end{array}\end{array}\end{eqnarray} \tag{ B5 }$

To this precision, the answer is the same if the calculation is performed again for an energy detector with the zero-point provided by VOSA. We used the zero-points provided by VOSA (as defined in Equation (B2)) to calculate synthetic photometry for Altair for standard filters used in the literature (Table 4 and Figure 20), in the energy-detector approximation. We find the average of the difference between our synthetic magnitudes and literature magnitudes to be 〈m_syn − m_lit〉 = +0.17, with a standard deviation of 0.10. We subtract this offset from the result in Equation (B5), and consider the standard deviation to be the error in the magnitude. This yields our adopted value of the absolute magnitude in this bandpass:

$\begin{eqnarray}&&{M}_{{\rm{A}}{\rm{l}}{\rm{t}}{\rm{a}}{\rm{i}}{\rm{r}}}=1.70\pm \mathrm{0.10.}\end{eqnarray} \tag{ B6 }$

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We also considered the possibility of color error in the measurement of the magnitude difference Δm between a host star and a companion, because we measure this difference between objects with different spectra from behind a terrestrial atmosphere in which transmission is a strong function of wavelength.

To quantify the error we should expect, we used an atmospheric transmission model from ATRAN (Lord 1992), corresponding to the latitude and altitude of the LBT, water vapor of 11 mm H₂O, and a zenith angle of 30° (Figure 21). Together with the Kurucz model spectrum for Altair, we tested blackbodies ranging from T = 200 K to T = 2800 K to mimic companion planet spectra, and found the magnitude error in Δm to always have an absolute value <0.01 mag, which is negligible for our purposes.

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In the past, much of the high-contrast literature has made use of "5σ" contrast curves, which are based on taking the level of the standard deviation of the noise at a given radius and multiplying it by five to obtain the amplitude of a signal where the false positive fraction (FPF) is expected to be ∼3 × 10⁻⁷ for Gaussian-distributed noise. However, this measure implies that the amplitude of the true positive fraction (TPF) is fixed at 0.5, since noise will conspire to make the other half of true signals fall below the threshold (Jensen-Clem et al. 2017).

Furthermore, the decreasing number of degrees of freedom at small radii—due to the decreasing number of FWHMs that can be fit onto a given annulus, like beads on a shrinking necklace—makes empirical approximations of the parent noise distribution increasingly coarse (Mawet et al. 2014). If we are willing to be flexible with the FPF, we can generate a more informative contrast curve that takes into account shrinking degrees of freedom at smaller angles from the host star.

The following is based on some of the methodologies and notation conventions from Mawet et al. (2014), Ruane et al. (2017), and Stone et al. (2018). We start by making the approximating assumption that at each radius from the host star the amplitudes of the parent speckle population are indeed Gaussian-distributed ${ \mathcal N }(\mu ,\sigma )$. In that case, we still do not know what the true parameters μ and σ of the Gaussian are, based on empirical measurements. We must resort to modeling the probability density distribution of the rescaled variable $t\equiv \tfrac{\bar{x}-\mu }{\sigma /\sqrt{n}}$ as a t-distribution, where n is the empirical sample size and $\bar{x}$ is the empirical average (Student 1908):

$\begin{eqnarray}&&{P}_{t}(t)=\displaystyle \frac{{\rm{\Gamma }}(n/2)}{\sqrt{\pi (n-1)}{\rm{\Gamma }}\left(\tfrac{n-1}{2}\right)}{\left(1+\displaystyle \frac{{t}^{2}}{n-1}\right)}^{-n/2}.\end{eqnarray} \tag{ C1 }$

This distribution has degrees of freedom ν = n − 1, and the gamma function ${\rm{\Gamma }}(z)\equiv {\int }_{0}^{\infty }{e}^{-t}{t}^{z-1}{dt}$ for real arguments z > 0. When comparing the means of two populations where the true variances are assumed to be the same, the t-statistic becomes

$\begin{eqnarray}&&t=\displaystyle \frac{{\bar{x}}_{1}-{\bar{x}}_{2}}{{s}_{\mathrm{1,2}}\sqrt{\tfrac{1}{{n}_{1}}+\tfrac{1}{{n}_{2}}}}=\displaystyle \frac{{x}_{1}-{\bar{x}}_{2}}{{s}_{2}\sqrt{1+\tfrac{1}{{n}_{2}}}}\end{eqnarray} \tag{ C2 }$

with variables defined in Table 5. (See expanded versions of this expression in, for example, Section 11.3.2 in Martin (1971).) The amplitude of a companion that fulfills our TPF and FPF criteria is ${\mu }_{c}={x}_{1}-{\bar{x}}_{2}\approx {x}_{1}$.

Table 5. Defined Quantities for Calculating the λ/D Contrast Curve

Parameter	Description	Units
${\bar{x}}_{1}={x}_{1}$	Fake companion amplitude, "averaged" over one value	Counts
${\bar{x}}_{2}\approx 0$	Amplitude of the noise ring, averaged over NFWHM,r − 1, the rounded floor number of FWHM that can fit in an annulus at a given radius of the host star, minus 1 to remove the location of the fake companion	Counts
s1,2 = s2	Empirical standard deviation of pooled samples 1 and 2.	Counts
	Since sample 1 just has one sample, this is the standard deviation of the amplitudes of the noise necklace beads
s2	Empirical standard deviation of the amplitudes of the noise necklace beads	Counts
A5	Amplitudes of fake injected planets, following smoothing by convolution with a Gaussian	Counts

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We apply the constraint that we are only willing to tolerate a total number of false positive detections ${N}_{\mathrm{FP}}^{(\mathrm{tot})}=0.01$ in a data set that ranges in angle out to a maximum integer radius of ${R}_{\max }=20$ in units of FWHM. Then at each radius there is a constant number of false positives ${N}_{{\rm{FP}},r}={N}_{{\rm{FP}}}^{({\rm{tot}})}/{R}_{max}=5\times {10}^{-4}$, and the FPF is a function of radius:

$\begin{eqnarray}&&{\rm{FPF}}(r)=\displaystyle \frac{{N}_{{\rm{FP}},r}}{2\pi r}.\end{eqnarray} \tag{ C3 }$

We want to find the threshold t = τ at each radius from the host star that satisfies the following constraints on the FPF and TPF of detections under the hypotheses H₀ (there is no companion) and H₁ (there is a companion):

$\begin{eqnarray}\begin{array}{c}\begin{array}{rcl}{\rm{FPF}}(r) & = & {\displaystyle \int }_{\tau }^{\infty }{P}_{t}(t| {H}_{0}){dt},\\ {\rm{TPF}} & = & 0.95={\displaystyle \int }_{{\mu }_{c}-\tau }^{\infty }{P}_{t}(t| {H}_{1}){dt}.\end{array}\end{array}\end{eqnarray} \tag{ C4 }$

To make the expressions simultaneously invertible to find the companion amplitude μ_c that satisfies both expressions, the lower bound of the TPF integral is the offset μ_c − τ, so that we can find that bound by using the CDF of the t-distribution at μ_c − τ. We then solve for the bounds by using the inverse of the cumulative distribution function, the percentage point function, of the t-distribution:

$\begin{eqnarray}&&{\mu }_{c}={\mathrm{ppf}}_{t,\tau }(1-\mathrm{FPF}(r))+{\mathrm{ppf}}_{t,{\mu }_{c}-\tau }(\mathrm{TPF}=0.95).\end{eqnarray} \tag{ C5 }$

This threshold is still in t-space and needs to be scaled to contrast, based on the results of the fake planet injections. Those injected planets converged—following a smoothing to remove pixel-to-pixel noise—on amplitudes A₅ corresponding to S/N = 5. Convergence on this value folds in throughput effects. The empirical noise is found from those rescaled injected companions as s₂ = A₅/5.

From Equation (C2) we have, for a given radius from the host star, the linear contrast F of a companion

$\begin{eqnarray}&&F={\mu }_{c}{s}_{2}\sqrt{1+\displaystyle \frac{1}{{n}_{2}}}.\end{eqnarray} \tag{ C6 }$

This corresponds to a contrast curve corrected for diminishing degrees of freedom at small angles, and which satisfies our constraints on the FPF and TPF. The λ/D contrast curve in Figure 13 is the magnitude equivalent of this, namely ${\rm{\Delta }}m=-2.5{\mathrm{log}}_{10}(F)$.